Explain the “value” of the theorem: the third standard of good textbooks

This is my about Linear Algebra. Learn versus teaching Guide to the third article in the series. This series of articles aims to clarify the standard of a good textbook in my mind; and briefly explain my original intention of writing the series of lectures in Linear Algebra.
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Note. At the end of the article, there is a handout link related to this article. Welcome to download ~ (๑• . •๑)

In the opening of this series Explain the main line: the primary standard of a good textbook In this article, I mentioned that many students have two very strong confusions when they first studied the basic course of Linear Algebra:

  1. (origin) How did these concepts come from? [Click to view the second article of this series]
  2. (Value) What are these theorems used for?

Previous article I talked about the first question. Today we will talk about the second question.

“Definition” and “Theorem” can basically be regarded as “the basic material of mathematics textbooks” – “Definition” introduces the research object of the whole subject (such as “linear mapping” and “matrix” in “linear algebra”) or “research tools” (such as “polynomial” to “linear algebra”); The “theorem” refers to the connection between two concepts. (For example, “Hamilton-Cayley Theorem” refers to the relationship between “characteristic polynomial” and “zero polynomial”).

Once you have clarified the source and connotation of the definition, “What is the value of this theorem?” – This question has become the primary concern of learning.

When a person asks something “what is useful” – I think this is a question of its value – some teachers may think that this student is just a refined egoist who is quick and quick, but I feel that I can ask this question and show that the student is trying to find the connection between this new concept and other things. The “other things” that I am talking about include both the old knowledge already in the student’s cognitive scope; Students try to understand new knowledge.

all in all, When a teacher tells new knowledge and faces the question of the student “what is the use of this thing”, if he leads well, he can either help the student to firmly implant this new concept into his old knowledge system to strengthen Its foundation; either can guide students to explore more new knowledge, leaving a curiosity and desire for his future study.

Although there are many math teachers who can explain the knowledge in any place and in any era in the world, it is frustrating to be able to explain carefully to the students (why, why should this concept be The teacher who defines this?” and “What is the use of this theorem?” is forever a handful.

even, At some point, some teachers who are unable to explain to students what certain important knowledge is “useful” (perhaps their ability is not enough to explain these problems) are also trying to invoke the so-called “useless use” in certain philosophies. The mystic methodology hides the incompetence of teaching itself—a situation that is extremely regrettable. It uses the most savage way to directly annihilate the original motivation of students’ knowledge, and also makes the title of “educator” shameful.

Those who can overcome this difficulty and try to explain the origins of the knowledge to the students and the teachers who use them: their lectures may be a classic of handed down; the students they have cultivated are expected to become a generation.

For example, every student studying analytics will listen to the four lectures that Professor Stein taught at Princeton University to teach mathematics undergraduate study analytics – Fourier Analysis \ Complex Analysis \ Real Analysis \ Functional Analysis — these books are very It perfectly explains the important criteria for a good teacher to lecture: to explain the origin of the concept (why we define it) and to explain the value of the theorem (what is the use of this theorem).

At least, when I was in the introductory course of PDE in the third year of college, I heard the concept of the so-called Kernels and Convolutions when I talked to the teacher about Laplace & Fourier Transform. In the fog, when reading Stein’s book, it was “Loosely speaking, convolutions.” Correspondence to weighted averages. “Let me wake up as a dream; when I learn the numerical solution of differential equations, I hear the teacher speak Delta Functions. I feel like I don’t understand, but Professor Stein is talking about the three conditions of the “Good Kernels” concept. “Why do you define this?” A footnote at the time made me understand what the Delta Functions are for!

Therefore, Teacher Stein can train students like Professor Tao Zhexuan, and maybe it has some truth.

Go back to what I am going to do.

When I first entered the university and became a mathematics student, I learned the basic courses of Mathematical Analysis and Linear Algebra. I first encountered the large-scale matrix operation of “similar diagonalization”. The question I most want to know is: We have to work so hard to make the square matrix similarly diagonalized. What is the significance of the similar diagonal type? What is the status of the condition that the square matrix can be similarly diagonalized throughout the course?

You all know, I am trying to write a “Linear Algebra” instructional for the students in the lower grades of the university. The purpose of this course is to use the basic course of “Linear Algebra” as a carrier to deeply practice my standard for “a good textbook.” “Understanding” After all, there are too many theorists in this era and talking about each other. If their own ideas cannot be practiced in the teachings, then this set of ideas will certainly be questioned.

in Previous handout (extract password: 0dbu) In this, we talked about the sources of important concepts such as “linear space, linear mapping, matrices, eigenvalues, and eigenvectors.” In the next third lecture, we will begin with the “invariant subspace” and “feature vector” to explain the value of an important topic in the course “Linear Algebra” – the discrimination of similar diagonals:

 link: http://pan.baidu.com/s/1o8mK61s Password: 92ni

Similarly, I also equipped the lecture with a Live. “Line Selection: The Value of Similar Diagonal Types” Thank you for your support. After joining Live, you can also find a communication group in it to facilitate the discussion and discuss the problems you encountered in the process of “Linear Algebra” in the group. There will also be irregular welfare sent within the group.